# Exercise ID85

Algebra → Roots → Complex exercises
[Level: ] [Number of helps: 1] [Number of pictures: 0] [Number of steps: 21] [Number of characters: 0]

Simplify the following expression:

$\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}}{x-\sqrt{xy}}+\frac{\sqrt{y}}{x+\sqrt{xy}}\right)$

$\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}}{x-\sqrt{xy}}+\frac{\sqrt{y}}{x+\sqrt{xy}}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}\left(x+\sqrt{xy}\right)+\sqrt{y}\left(x-\sqrt{xy}\right)}{\left(x-\sqrt{xy}\right)\left(x+\sqrt{xy}\right)}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{x\sqrt{y}+y\sqrt{x}+x\sqrt{y}-y\sqrt{x}}{{x}^{2}-xy}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{2x\sqrt{y}}{{x}^{2}-xy}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{2x\sqrt{y}}{x\left(x-y\right)}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{2\sqrt{y}}{x-y}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{x}\sqrt{y}}\left(\frac{2\sqrt{y}}{x-y}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}}\left(\frac{1}{x-y}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}}\left(\frac{1}{x-y}\right)$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}}\text{}·\frac{1}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{1}{\sqrt{x}}\text{}·\frac{1}{\sqrt{x}+\sqrt{y}}$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{1}{\sqrt{x}}\text{}·\frac{1}{\sqrt{x}+\sqrt{y}}$ $=\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{1}{x+\sqrt{xy}}$ $=\frac{\sqrt{x}+\sqrt{y}-1+1}{x+\sqrt{xy}}$ $=\frac{\sqrt{x}+\sqrt{y}}{x+\sqrt{xy}}$ $=\frac{\sqrt{x}+\sqrt{y}}{x+\sqrt{xy}}·\frac{x-\sqrt{xy}}{x-\sqrt{xy}}$ $=\frac{\sqrt{x}+\sqrt{y}}{{x}{+}\sqrt{xy}}·\frac{x-\sqrt{xy}}{{x}{-}\sqrt{xy}}$ $=\frac{x\sqrt{x}{+}{x}\sqrt{y}{-}{x}\sqrt{y}-y\sqrt{x}}{{{x}}^{2}{-}{x}{y}}$ $=\frac{x\sqrt{x}-y\sqrt{x}}{{x}^{2}-xy}$ $=\frac{\sqrt{x}{\left(}{x}{-}{y}{\right)}}{x{\left(}{x}{-}{y}{\right)}}$ $=\frac{\sqrt{x}}{x}$
HELP AVAILABLE!
$\frac{\sqrt{x}+\sqrt{y}-1}{x+\sqrt{xy}}+\frac{\sqrt{x}-\sqrt{y}}{2\sqrt{xy}}\left(\frac{\sqrt{y}}{x-\sqrt{xy}}+\frac{\sqrt{y}}{x+\sqrt{xy}}\right)=\frac{\sqrt{x}}{x}$

${a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right)$